On superlinear elliptic problems with nonlinearities interacting only with higher eigenvalues (Q1120727)

There is given a bounded smooth domain \(\Omega \subset R^ N\), \(N\geq 2\). The author considers the parametric superlinear elliptic boundary value problem \[ (*)\quad -\Delta u=\lambda u+f(x,u)+t\phi +h\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] where \(\lambda_ k<\lambda <\lambda_{k+1}\), \(\lambda_ k\) and \(\lambda_{k+1}\) are two consecutive eigenvalues of \((-\Delta,H^ 1_ 0(\Omega))\), t is a real parameter, \(\phi >0\) is an eigenfunction associated with the first eigenvalue of \[ (-\Delta,H^ 1_ 0(\Omega)),\quad h\in C^{\mu}({\bar \Omega}),\quad 0<\mu \leq 1,\quad \int_{\Omega}h\phi dx=0. \] It is shown that under certain assumptions on the nonlinear term f, there exists a parameter \(t_ 0>0\) such that, for all \(t\geq t_ 0\) problem (*) has at least two solutions.